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Recently, authors have proposed under-approximate logics for reasoning aboutprograms. So far, all such logics have been confined to reasoning aboutindividual program behaviours. Yet there exist many over-approximate relationallogics for reasoning about pairs of programs and relating their behaviours. Wepresent the first under-approximate relational logic, for the simple imperativelanguage IMP. We prove our logic is both sound and complete. Additionally, weshow how reasoning in this logic can be decomposed into non-relationalreasoning in an under-approximate Hoare logic, mirroring Beringer's result forover-approximate relational logics. We illustrate the application of our logicon some small examples in which we provably demonstrate the presence ofinsecurity.

An Under-Approximate Relational Logic: Heralding Logics of Insecurity, Incorrect Implementation & More